Numerical Methods in Quantum Mechanics: Analyzing the Schrödinger Equation

For this study, the shooting method and leapfrog method were utilized for the time-independent Schrödinger equation and time-dependent Schrödinger equation respectively.

Time-Independent Schrödinger Equation: The Shooting Method

The first step is to convert the boundary value problem into an equivalent initial value problem. The boundary condition is given that wave function equal to zero at the walls of the box. We then iterate that guess undefined values of the energy of the particle inside of the box. Assuming that potential outside the box is infinite.

We begin at the initial energy and gradually increase or decrease the value until we want our end of the integration to hit the target.

By taking the second derivative of finite difference, we can derive the following equation:

-\frac{\hbar^2}{2m}\frac{d^2\psi}{{\rm dx}^2}\approx\frac{\hbar^2}{2m}[\frac{\psi_{n+1}+\psi_{n-1}-2\psi_n}{{(\Delta x)}^2}]≈(E-Vn)ψn

For convenience, we employ units in which \hbar=1\ and\ m=1.

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\ The equation can be therefore rewritten as

\psi_{n+1}=2\psi_n{\ -\ \psi}_{n-1}-\ 2{(\Delta x)}^2 (E-V_n){\ \psi}_n

We start with the particle in a box problem. Assuming that particle is confined in a x=\pmL of a box. Since A=L^{-\frac{1}{2}} is required for the wave functions to be properly normalized. A is therefore equivalent to 0.707. The time-independent Schrödinger equation then becomes

-\frac{\hbar}{2m}\frac{d^2\psi}{dx^2}\ =\ E\ \psi

We derive the energy of the particle by using the general solution \psi=A\ exp(kxj).

Hence E\ =\frac{\hbar^2k^2}{2m}

Where \hbar=1,\ m=1, wave vector \ k=2\pi/\lambda,

Parity \mathbf{k} E (energy of the particle) p(particle’s momentum)

Table 1. Simulation and Calculation Results

This data illustrates that the higher discrete energy levels, the higher the energy, which is a common property in all quantum systems. Here we also see that the particle’s momentum is in a linear relation with wave vector k.

The time-dependent Schrödinger equation for a particle in three dimensions is given by

-\frac{\hbar}{2m}\nabla^2\psi+V(\hvec{r})\psi\ =\ \hbar\ \frac{\partial\psi}{\partialt}i

The time-dependent Schrödinger equation describes how a wave function of a quantum evolves. The wave function is a very important concept of quantum mechanics and is a tool used to describe the state of the quantum state. There are several important mysteries related to quantum mechanics related to wave functions. The Schrödinger equation can describe the behaviour of wave functions.

\psi(x,t)=R(x,t)+jI(x,t)

It can be divided into two-part

\frac{\partialR}{\partialt}=[-\frac{1}{2}\ \frac{\partial^2}{\partialx^2}+V(x)\ ] I

\frac{\partialI}{\partialx}=[\ \frac{1}{2}\ \frac{\partial^2}{\partialx^2}-V(x)\ ] R

For these equations, the parameters of h and m are assumed to be unity. This is a common approach to these types of mathematical problems.

Here, we used a numerical approach to solve the Time-Dependent problem. Leapfrog integral method is a simple method to integrate differential equations, especially in the case of dynamic systems.

The leapfrog integration method is equivalent to calculating positions and velocities at alternate time points, and they are staggered in time, so they 'jump' over each other. For example, the position is a one-time step and the speed is an integer plus half the time step.

The onset of simulation, wave packet was set propagating on a flat surface. We generated an animation of the motion of a wave packet, the propagating of the wave packet will be therefore seen clearly that spreading out over time by using the program of leapfrog.

Influence of K__o on Wavefunction

K_0 is wave vector of wave function. This result shows that the linear relation between k_0 and velocity. Wave packet spreading out over time. This result show the width of wave packet is proportional to time. While, the length of wave packet is inversely proportional to time.

Table 2. The Value of k0 is Proportional to Velocity

There is one more observation of the reaction of wave packet through the potential barrier the wave packet be set travelling through various potential barriers in this simulation, such as linear potential, harmonic potential, and so on. According to wave packet, the amplitude of potential energy, the thickness of barriers, and other physical properties are modified, and characteristics of wave packet also vary accordingly

An important result of the work is to use numerical analysis to study the possibility of quantum tunneling by changing the magnitude of the potential energy and the thickness of the barrier.

Next, we consider another scenario where the length of the barrier is even shorter. In this case, the wave function does not have much distance to decay inside the barrier, we, therefore, have a larger amplitude for the portion of the wave function that exits the barrier.

What is significant is that with a shorter barrier, the particle has a greater probability of passing through. Also, it illustrates that the particle has a lower probability of bouncing off the barrier, which is represented by a smaller amplitude for the reflected wave. As we increase the length of the barrier, the particle is less likely to pass through, and more likely to bounce off.

Conclusion

The numerical methods applied to the Schrödinger equation illuminate the intricate behaviors of quantum systems. Through the shooting method, we discern the quantization of energy levels in a potential well. The leapfrog method further enables the visualization of dynamic quantum phenomena, such as tunneling and wave packet dispersion. These numerical approaches not only validate theoretical predictions but also offer invaluable tools for exploring the quantum realm, highlighting the symbiosis between computational physics and quantum mechanics.